A solid cone of mass 'm' falls from a height 'h' and penetrates into sand. The resistance force of the sand is given by $R = kx^{2}$ . If the cone penetrates upto a distance d where d<l , then find the value of 'k'

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TestHub · Accepted Answer

Apply the work-energy theorem over the whole motion - free fall through height h and penetration to depth d:

Loss of gravitational potential energy = mg(h + d).

Work done against the sand resistance:

$W = \int_{0}^{d} kx^{2} dx = \frac{kd ^{3}}{3}$

$mg (h + d) = \frac{kd ^{3}}{3}$

$\Rightarrow k = \frac{3 mg}{d ^{3}} (h + d)$

Correct option: (4).

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